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2 Applicationes Mathematicae

2 2011

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Convergence domains under Zabrejko-Zinčenko conditions using recurrent functions

100%

Argyros I. , Hilout S.

nr 2

193-209

We provide a semilocal convergence analysis for Newton-type methods using our idea of recurrent functions in a Banach space setting. We use Zabrejko-Zinčenko conditions. In particular, we show that the convergence domains given before can be extended under the same computational cost. Numerical examples are also provided to show that we can solve equations in cases not covered before.

On the convergence of Newton's method under ω*-conditioned second derivative

100%

Argyros I. , Hilout S.

nr 3

341-355

We provide a new semilocal result for the quadratic convergence of Newton's method under ω*-conditioned second Fréchet derivative on a Banach space. This way we can handle equations where the usual Lipschitz-type conditions are not verifiable. An application involving nonlinear integral equations and two boundary value problems is provided. It turns out that a similar result using ω-conditioned hypotheses can provide usable error estimates indicating only linear convergence for Newton's method.